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Fabrice Baudoin

Publications and source records attributed to Fabrice Baudoin.

At least 37 records · Page 2Linked to original sources
arXiv2024

On the law of the index of Brownian loops related to the Hopf and anti-de Sitter fibrations

We give explicit formulas and asymptotics for the distribution of the index of the Brownian loop in the following geometrical settings: the complex projective line from which two points have been removed; the complex hyperbolic line from which one point has been removed; the odd dimensional spheres from which a great hypersphere has been removed; and the complex anti-de Sitter spaces. Our analysis is based on the geometry of the Hopf and anti-de Sitter fibrations, and on the relationship between winding and area forms.

math.PR↗
arXiv2024

Korevaar-Schoen-Sobolev spaces and critical exponents in metric measure spaces

We present developments in the theory of Korevaar-Schoen-Sobolev spaces on metric measure spaces. While this theory coincides with those of Cheeger and Shanmugalingam if the space is doubling and satisfies a Poincaré inequality, it offers new perspectives in the context of fractals for which the approach by weak upper gradients is inadequate.

math.MG↗
arXiv2024

Heat kernel gradient estimates for the Vicsek set

We prove pointwise and $L^p$ gradient estimates for the heat kernel on the bounded and unbounded Vicsek set and applications to Sobolev inequalities are given. We also define a Hodge semigroup in that setting and prove estimates for its kernel.

math.AP↗
arXiv2024

Fractional stable random fields on the Sierpiński gasket

We define and study fractional stable random fields on the Sierpiński gasket. Such fields are formally defined as $(-Δ)^{-s} W_{K,α}$, where $Δ$ is the Laplace operator on the gasket and $W_{K,α}$ is a stable random measure. Both Neumann and Dirichlet boundary conditions for $Δ$ are considered. Sample paths regularity and scaling properties are obtained. The techniques we develop are general and extend to the more general setting of the Barlow fractional spaces.

math.PR↗
arXiv2024

The indifference value of the weak information

We propose indifference pricing to estimate the value of the weak information. Our framework allows for tractability, quantifying the amount of additional information, and permits the description of the smallness and the stability with respect to small perturbations of the weak information. We provide sharp conditions for the stability with counterexamples. The results rely on a theorem of independent interest on the stability of the optimal investment problem with respect to small changes in the physical probability measure. We also investigate contingent claims that are indifference price invariant with respect to changes in weak information. We show that, in incomplete models, the class of information-invariant claims includes the replicable claims, and it can be strictly bigger. In particular, in complete models, all contingent claims are information invariant. We augment the results with examples and counterexamples.

q-fin.MF↗
arXiv2024

Oscillations of BV measures on nested fractals

Motivated by recent developments in the theory of bounded variation functions on nested fractals, this paper studies the exact asymptotics of functionals related to the total variation measure associated with unions of $n$-complexes. The oscillatory behavior observed implies the non-uniqueness of BV measures in this setting.

math.MG↗
arXiv2024

Korevaar-Schoen $p$-energies and their $Γ$-limits on Cheeger spaces

This paper studies properties of $Γ$-limits of Korevaar-Schoen $p$-energies on a Cheeger space. When $p>1$, this kind of limit provides a natural $p$-energy form that can be used to define a $p$-Laplacian, and whose domain is the Newtonian Sobolev space $N^{1,p}$. When $p=1$, the limit can be interpreted as a total variation functional whose domain is the space of BV functions. When the underlying space is compact, the $Γ$-convergence of the $p$-energies is improved to Mosco convergence for every $p \ge 1$.

math.FA↗
arXiv2024

Parabolic Anderson model in bounded domains of recurrent metric measure spaces

A metric measure space equipped with a Dirichlet form is called recurrent if its Hausdorff dimension is less than its walk dimension. In bounded domains of such spaces we study the parabolic Anderson models \[ \partial_{t} u(t,x) = Δu(t,x) + βu(t,x) \, \dot{W}_α(t,x) \] where the noise $W_α$ is white in time and colored in space when $α>0$ while for $α=0$ it is also white in space. Both Dirichlet and Neumann boundary conditions are considered. Besides proving existence and uniqueness in the Itô sense we also get precise $L^p$ estimates for the moments and intermittency properties of the solution as a consequence. Our study reveals new exponents which are intrinsically associated to the geometry of the underlying space and the results for instance apply in metric graphs or fractals like the Sierpiński gasket for which we prove scaling invariance properties of the models.

math.PR↗
arXiv2023

Functional inequalities for a family of infinite-dimensional diffusions with degenerate noise

For a family of infinite-dimensional diffusions with degenerate noise, we develop a modified $Γ$ calculus on finite-dimensional projections of the equation in order to produce explicit functional inequalities that can be scaled to infinite dimensions. The choice of our $Γ$ operator appears canonical in our context, as the estimates depend only on the induced control distance. We apply the general analysis to a number of examples, exploring implications for quasi-invariance and uniqueness of stationary distributions.

math.PR↗
arXiv2023

Stochastic areas, Horizontal Brownian Motions, and Hypoelliptic Heat Kernels

The monograph is devoted to the study of stochastic area functionals of Brownian motions and of the associated heat kernels on Lie groups and Riemannian manifolds. It is essentially self-contained and as such can serve as a textbook on the theory of Brownian motions and horizontal Brownian motions on manifolds. Emphasis is put on concrete examples which allows us to concretely illustrate the rich and deep interactions between stochastic calculus, Riemannian and sub-Riemannian geometry, the theory of complex and quaternionic symmetric spaces and random matrices.

math.PR↗
arXiv2023

$L^p$-Poincaré inequalities on nested fractals

We prove on some nested fractals scale invariant $L^p$-Poincaré inequalities on metric balls in the range $1 \le p \le 2$. Our proof is based on the development of the local $L^p$-theory of Korevaar-Schoen-Sobolev spaces on fractals using heat kernel methods. Applications to scale invariant Sobolev inequalities and to the study of maximal functions and Hajłasz-Sobolev spaces on fractals are given. Results are illustrated and further developed in the case of the Vicsek set.

math.FA↗
arXiv2023

Dirichlet forms on metric measure spaces as Mosco limits of Korevaar-Schoen energies

This paper establishes sufficient general conditions for the existence of Mosco limits of Korevaar-Schoen $L^2$ energies, first in the context of Cheeger spaces and then in the context of fractal-like spaces with walk dimension greater than 2. Among the ingredients, a new Rellich-Kondrachov type theorem for Korevaar-Schoen-Sobolev spaces is of independent interest.

math.FA↗
arXiv2022

Variations of the sub-Riemannian distance on Sasakian manifolds with applications to coupling

On Sasakian manifolds with their naturally occurring sub-Riemannian structure, we consider parallel and mirror maps along geodesics of a taming Riemannian metric. We show that these transport maps have well-defined limits outside the sub-Riemannian cut-locus. Such maps are not related to parallel transport with respect to any connection. We use this map to obtain bounds on the second derivative of the sub-Riemannian distance. As an application, we get some preliminary result on couplings of sub-Riemannian Brownian motions.

math.DG↗
arXiv2022

H-type foliations

With a view toward sub-Riemannian geometry, we introduce and study H-type foliations. These structures are natural generalizations of K-contact geometries which encompass as special cases K-contact manifolds, twistor spaces, 3K contact manifolds and H-type groups. Under an horizontal Ricci curvature lower bound, we prove on those structures sub-Riemannian diameter upper bounds and first eigenvalue estimates for the sub-Laplacian. Then, using a result by Moroianu-Semmelmann, we classify the H-type foliations that carry a parallel horizontal Clifford structure. Finally, we prove an horizontal Einstein property and compute the horizontal Ricci curvature of those spaces in codimension more than 2.

math.DG↗
arXiv2022

Sobolev spaces and Poincaré inequalities on the Vicsek fractal

In this paper we prove that several natural approaches to Sobolev spaces coincide on the Vicsek fractal. More precisely, we show that the metric approach of Korevaar-Schoen, the approach by limit of discrete $p$-energies and the approach by limit of Sobolev spaces on cable systems all yield the same functional space with equivalent norms for $p>1$. As a consequence we prove that the Sobolev spaces form a real interpolation scale. We also obtain $L^p$-Poincaré inequalities for all values of $p \ge 1$.

math.MG↗
arXiv2022

Besov class via heat semigroup on Dirichlet spaces III: BV functions and sub-Gaussian heat kernel estimates

With a view toward fractal spaces, by using a Korevaar-Schoen space approach, we introduce the class of bounded variation (BV) functions in a general framework of strongly local Dirichlet spaces with a heat kernel satisfying sub-Gaussian estimates. Under a weak Bakry-Émery curvature type condition, which is new in this setting, this BV class is identified with a heat semigroup based Besov class. As a consequence of this identification, properties of BV functions and associated BV measures are studied in detail. In particular, we prove co-area formulas, global $L^1$ Sobolev embeddings and isoperimetric inequalities. It is shown that for nested fractals or their direct products the BV class we define is dense in $L^1$. The examples of the unbounded Vicsek set, unbounded Sierpinski gasket and unbounded Sierpinski carpet are discussed.

math.MG↗
arXiv2022

Parabolic Anderson model on Heisenberg groups: the Itô setting

In this note we focus our attention on a stochastic heat equation defined on the Heisenberg group $\mathbf{H}^{n}$ of order $n$. This equation is written as $\partial_t u=\frac{1}{2}Δu+u\dot{W}_α$, where $Δ$ is the hypoelliptic Laplacian on $\mathbf{H}^{n}$ and $\{\dot{W}_α; α>0\}$ is a family of Gaussian space-time noises which are white in time and have a covariance structure generated by $(-Δ)^{-α}$ in space. Our aim is threefold: (i) Give a proper description of the noise $W_α$; (ii) Prove that one can solve the stochastic heat equation in the Itô sense as soon as $α>\frac{n}{2}$; (iii) Give some basic moment estimates for the solution $u(t,x)$.

math.PR↗